Abstract:
The standard version of formal axiomatic method stemming from Hilbert [Hilbert (1899)] and recently defended by Hintikka [Hintikka (2011)] is not fully adequate to the recent successful practice of axiomatizing mathematical theories. In particular, the axiomatic architecture of Homotopy Type theory (HoTT) [Voevodsky et. al. 2013] does not quite fit the standard Hilbertian pattern of axiomatic theory. At the same time HoTT and some other recent theories fall under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert and Bernays [Hilbert&Bernays (1934-1939)] “genetic” or “constructive” (interchangeably) and demonstrate it using the Classical example of the First Book of Euclid’s “Elements”. On the basis of these modern and ancient examples I claim that Hintikka’s semantic-oriented formal axiomatic method is not self-sustained but requires a support of some more basic constructive method. I provide an independent epistemological grounding for this claim by showing the need to complement Hintikka’s account of axiomatic method with a constructive notion of formal semantics.

September 14, 2014

Summary: Knowledge Representation is a rapidly developing area of Artificial Intelligence, which brings together computer scientists and philosophers. The purpose of this workshop is to explore some applications of Category theory and Ontologies in Knowledge Representation.

August 15, 2014

The formal axiomatic method popularized by Hilbert and recently defended by Hintikka is not fully adequate to the recent practice of axiomatizing mathematical theories. The axiomatic architecture of Topos theory and Homotopy type theory do not fit the pattern of the formal axiomatic theory in the standard sense of the word. However these theories fall under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert constructive. I show that the formal axiomatic method always requires a support of some more basic constructive method.